Similar to the case of amplitude birefringence, the velocity birefringence due to parity violation can also be tested by comparing the modified waveform (REF ) with the GW strain data from the GW detectors. Based one this, the tests of velocity birefringence have been carried out through full Bayesian parameter estimations on the GW events observed by the LIGO/Virgo/KAGRA detectors in a series of papers [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. Here we would like to derive the constraint on coefficient \({\cal W}_1 - {\cal G}_1\)  by directly using the posterior samples obtained in [3]} for testing the velocity birefringent effect with 94 GW events reported by the 4th-open GW catalog (4-OGC) [10]}.
The data for these posterior samples are downloaded from [11]}. With these data, one can derive the posterior distributions of \({\cal W}_1 - {\cal G}_1\)  from the posterior distributions of the sampled parameter \(M_{\rm PV}^{-1}\)  for each GW event. In Fig. REF , we present briefly the posterior distributions of \(|{\cal W}_1 - {\cal G}_1|\)  for the 92 GW events analyzed Here we exclude the two events GW190521 and GW191109, since their posterior samples show intriguing non-zero results for velocity birefringence. Some possibilities produce such signature have also been explored briefly in [3]}.. Note that here we have treated the coefficient \({\cal W}_1 - {\cal G}_1\)  as constant as well. From this figure, we find that the posterior distributions of \(|{\cal W}_1 - {\cal G}_1|\)  with 90% confidence intervals are consistent with the GR value \({\cal W}_1 - {\cal G}_1=0\)  for all the 92 GW events. By considering \(|{\cal W}_1 - {\cal G}_1|\)  as a universal parameter, we also present its upper bound from the combined posterior probability distributions in Fig. REF  (the vertical dash line), from which one is able to place a constraint on \(|{\cal W}_1 - {\cal G}_1|\)  as
\(|{\cal W}_1 - {\cal G}_1| < 4.4 \times 10^{-18} \; {\rm km}, \) 
